stability in odes: show that all solutions to the Lorenz equations enter region H less than or equal to and never leave it

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classical analysis and classical odes: limit of a Fourier coefficient of a non-negative periodic function in terms of its form $ L ^ 2 $

This question is motivated by the previous MO question: Show that $ ( sum_ {k = 1} ^ {n} x_ {k} cos {k}) ^ 2 + ( sum_ {k = 1} ^ {n} x_ {k} sin {k}) ^ 2 le (2+ frac {n} {4}) sum_ {k = 1} ^ {n} x ^ 2_ {k} $.
It is a clean asymptotic version of that question.

Leave $ f $ be a non-negative, periodic function with period $ 1 $and square integrable in $ { Bbb R} / { Bbb Z} $. It is true that
$$
| { widehat f} (1) | ^ 2 = Big | int_0 ^ 1 f (x) e ^ {- 2 pi ix} dx Big | ^ 2 le frac 14 int_0 ^ 1 f (x) ^ 2 dx ?
$$

Equality is achieved, for example, when $ f (x) = max (0, cos (2 pi x)) $.

Note that $ | widehat f (1) | = | widehat f (-1) | $ and from $ f $ it is not negative $ | widehat f (1) | le widehat f (0) $. Thus
$$
int_0 ^ 1 f (x) ^ 2 dx = sum_n | widehat f (n) | ^ 2 ge 3 | widehat f (1) | ^ 2,
$$

so that the estimate is maintained with $ 1/3 $ instead of $ 1/4 $. There is a lot of room to improve this argument, and with a more careful application of Bessel's inequality I could get the constant $ 1/4 + 1/4 pi $. But the alleged inequality looks very clean, and I wonder if (i) is true!, (Ii) is known in some context, and (iii) (hopefully) has an elegant proof?

Classic analysis and classic odes: Who is the best poets in the world and the 10 best famous poets?

Here is a list of the 10 most famous English poets and writers of all time. Now we are going to discuss the 10 most famous poets and writers of all time.

World Poetry Day is celebrated every year on March 21, observing the sensitive words, rhymes and rhymes of poets from around the world. Held on March 21 of each year, World Poetry Day celebrates the delicate words, rhymes and rhythms of poets throughout the world. Celebrate poetry, an excellent representation of cultural and linguistic manifestations themselves.

Classic analysis and odes: mass of a bounded region

Does anyone know how to solve this? The region is really confusing me.

Leave $ T $ be the region delimited by $ (x, y, z) in mathbb {R} ^ 3 $
that satisfy the inequality:
$$
0 < sqrt {x ^ 2 + y ^ 2 + z ^ 2} <1- left | z right |
$$

Find the mass of $ T $, where the density is given by:
$$
rho left (x, y, z right) = left (x ^ 2 + y ^ 2 + z ^ 2 right) ^ {- frac {3} {4}}
$$

What I have done so far is:
$$
0 le sqrt {x ^ 2 + y ^ 2 + z ^ 2} le 1- left | z right | + Rightarrow left (x ^ 2 + y ^ 2 + z ^ 2 right) le left (1- left | z right | right) ^ 2
$$

$$
Rightarrow : x ^ 2 + y ^ 2 le 1-2 left | z right | + z ^ 2-z ^ 2 Rightarrow : x ^ 2 + y ^ 2-1 + le 2 left | z right |
$$

$$
Rightarrow : : left | z right | le frac {1} {2} left (1-x ^ 2-y ^ 2 right) Rightarrow : z le frac {1} {2} left (1-x ^ 2-y ^ 2 right) : vee : z ge frac {1} {2} left (x ^ 2 + y ^ 2-1 right)
$$

xy-plane gets that: $ y pm sqrt {1-x ^ 2} $ such that:
$$
int _ {- 1} ^ 1 int _ {- sqrt {1-x ^ 2}} ^ { sqrt {1-x ^ 2}} int _ { frac {1} {2} left (x ^ 2 + y ^ 2-1 right)} ^ { frac {1} {2} left (1-x ^ 2-y ^ 2 right) :} dzdydx : : :
$$

Is my solution correct so far? And what do I do now

differential equations: solve two more Odes

why Mathematica gave no output when iam tried to use d solve to solve both equations, any comments please

Θ (y_) = c1ⅇ ^ A1y + c2ⅇ ^ (- A1y); Θp (y _) = 1 / (s + ϵ) (c1ⅇ ^ A1y + c2ⅇ ^ (- A1y));
DSolve ({u & # 39; & # 39; (y) – (s + M) u (y) + ksα (up (y) -u (y)) + GrΘ (y) == 0,1 / β1 up & # 39; & # 39; (y) -s up (y) -α (up (y) -u (y)) + GrΘp (y) == 0}, {u, up}, y)

Classic analysis and odes: if a bivariate function of real value in the square of the unit is integrable along each line, is it integrable in the square?

No, consider the function
$$ g (x, y) = cases {0, y if $ (x, y) = (0,0) $ \
xy ^ 2 / (x ^ 2 + y ^ 6) and otherwise} $$

taken from Exercise 4.7 of Baby Rudin (3ed) through this publication of Math.SE. Along each line it is continuous and, therefore, Riemann integrable, but has no limits along the curve. $ x = y ^ 3 $, and an unlimited function cannot be integrated by Riemann.

Classic analysis and odes: an identity between two elliptical integrals

I would like a direct change of variable proof of identity.

$$ int_0 ^ ( arctan frac {sqrt {2}} { sqrt { sqrt {3}}) 1 / sqrt {1- frac {2+ sqrt {3}} {4} sin ^ 2 phi d phi} = int_0 ^ arctan frac {1} { sqrt { sqrt {3}} 1 / sqrt {1- frac {2+ sqrt {3}} {4} sin ^ 22 phi d phi} = ,. $$

I need it as part of a document on the Legendre test of the "third singular module."

Ordinary differential equations – Qualitative theory of odes. Show that each path in $ X $ is dense in $ X $ and $ alpha (x) = omega (x) $ for all $ x in X $.

Leave $ X $ be a compact set not empty and invariable to the flow determined by a $ C 1 $ vector field Suppose that $ X $ It is a minimal set.
Test it:

a) Each trajectory in $ X $ is dense in $ X $.

yes) $ alpha (x) = omega (x) $ for all $ x in X $.

Note that: α (x) is the limit α and ω (x) the limit ω of x.

Classic analysis and odes: an optimization problem for Schrodinger's one-dimensional operator

For a potential of the form $ V (x) = ax ^ 4 + bx ^ 2 $, where $, a, b> 0 $, consider Schrodinger's one-dimensional operator $ D = frac {d ^ 2} {dx ^ 2} + V $ with Dirichlet B.C in $ (- L, L) $ and denotes its first proper value by $ lambda (a, b) $.

Q1 Is $ lambda (a, b) $ a differentiable function in $ to $ and $ b $?

Q2 For which $ (a, b) $It is subject to $ a + b = 1 $, the function $ lambda $ Is it maximized / minimized?

Classic analysis and odes – Reference request: Lie product formula for vector fields

Leave $ K $ be a compact neighborhood in $ mathbb R ^ n $, $ Z = X + Y $ are (do not disappear if necessary) soft vector fields in $ K $. We denote by $ e ^ sZ p $ an integral curve of $ Z $ with starting point $ p = e 0Z p in K $ and a terminal point $ q = e ^ {tZ} p in K $. Then for any $ varepsilon> 0 $ there is a sufficiently large natural number $ N $ such that

$$ || (e ^ { frac {s} {N} X} e ^ { frac {s} {N} Y}) ^ Np – e ^ {sZ} p || < varepsilon $$

for all $ 0 <s <t $.

I am looking for a reference for this statement (preferably a textbook)